Numerische Mathematik

, Volume 105, Issue 3, pp 413–455 | Cite as

The tent transformation can improve the convergence rate of quasi-Monte Carlo algorithms using digital nets

  • Ligia L. Cristea
  • Josef Dick
  • Gunther Leobacher
  • Friedrich Pillichshammer


In this paper we investigate multivariate integration in reproducing kernel Sobolev spaces for which the second partial derivatives are square integrable. As quadrature points for our quasi-Monte Carlo algorithm we use digital (t,m,s)-nets over \(\mathbb{Z}_2\) which are randomly digitally shifted and then folded using the tent transformation. For this QMC algorithm we show that the root mean square worst-case error converges with order \(2^{m(-2+\varepsilon)}\) for any ɛ >  0, where 2 m is the number of points. A similar result for lattice rules has previously been shown by Hickernell.

Mathematics Subject Classification (1991)



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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Ligia L. Cristea
    • 1
  • Josef Dick
    • 2
  • Gunther Leobacher
    • 1
  • Friedrich Pillichshammer
    • 1
  1. 1.Institut für FinanzmathematikUniversität LinzLinzAustria
  2. 2.Division of Engineering, Science & TechnologyUNSW AsiaSingaporeSingapore

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